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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2020 Paper 1 19 a Solution by 
Codex 0 2026-10-18
Codex 0 2026-10-18In the Cartesian coordinate system, the Laplacian of a scalar field is the sum of its three unmixed second partial derivatives. Each squared coordinate here contributes one, soEquivalently and its divergence is three.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2020 Paper 1 18 b Solution by Codex 0 2026-10-18
Codex 0 2026-10-18A square homogeneous linear system has a nonzero solution exactly when its determinant vanishes. Expanding along the second row givesConsequentlyFor an explicit check, the null space contains at and at .
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2020 Paper 1 17 b Solution by Codex 0 2026-10-18
Codex 0 2026-10-18The product in the integrand simplifies to below its upper endpoint. With the integration by substitution , the limits become and zero and . ThereforeAlthough the tangent function itself is undefined at the upper endpoint, the product has a zero limit there, so its continuous extension, or its improper integral, gives this same result.
To obtain directly from a Riemann sum, divide into equal pieces of width and use right endpoints . The sum isThe integrand is continuous, so the limit of these Riemann sums is its definite integral, namely , agreeing with the antiderivative calculation.
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2020 Paper 1 15 b Solution by Codex 0 2026-10-18
Codex 0 2026-10-18For , the first two derivatives are and . Since ,The first three terms of the Taylor expansion, with , therefore giveFor a check on accuracy, is positive and decreasing on . The Taylor remainder is consequently positive and less than . The approximation is a slight underestimate.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
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